The neutron life cycle in a Pressurized Water Reactor (PWR) Part 2(M J Rhoades)

In part 1, we discussed neutrons and their interactions with matter, crossection of absorption,

neutron energies, neutron velocities, as well as use of a moderator for the Pressurized Water

Reactor (PWR). It is now time to move on to the actual life cycle of neutrons in the core and

discuss the various factors that affect the neutron population and thus the criticality of our core.

We may as well go on and dive into it by stating out with the definition of "K." K Is the core

multiplication factor. It is a description of what is happening to the neutron population in the

core at any given time. I guess I need to mention that there is two types of K you will hear about,

K∞, and Keff. I am only going to mention K∞, because it is just a hypothetical and means

nothing. It is based on a core of infinite size, or total neutron reflection with no neutron leakage.

It is only an exercise in mathematics to talk about it. Keff is the real world K and it is defined as a

ratio as follows:

Keff = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑖𝑛 𝑎 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑒𝑢𝑡𝑟𝑜𝑛𝑠 𝑖𝑛 𝑡𝑕𝑒 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 or

Keff = 𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒

𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑙𝑜𝑠𝑠 𝑟𝑎𝑡𝑒 or

Keff = 𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑖𝑛 𝑜𝑛𝑒 𝑔𝑒𝑛𝑒𝑟 𝑎𝑡𝑖𝑜𝑛

𝑁𝑒𝑢𝑡𝑟𝑜𝑛 𝑎𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 𝑖𝑛 𝑡𝑕𝑒 𝑝𝑟𝑜𝑐𝑒𝑒𝑑𝑖𝑛𝑔 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛

+ 𝑛𝑒𝑢𝑡𝑟𝑜𝑛 𝑙𝑒𝑎𝑘𝑎𝑔𝑒 𝑖𝑛 𝑡𝑕𝑒

𝑝𝑟𝑜𝑐𝑒𝑑𝑖𝑛𝑔 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛

All this means is, that if there is a self sustaining chain reaction then Keff = 1, and the neutron

population is not increasing or decreasing. This is the definition of reactor criticality or the

reactor is critical. Keff is dependent on six factors, which when put together is called the six

factor formula. The formula uses six symbols and is written as follows:

Keff = ɛ Lf p Lt 𝑓 η Where: Keff = Core multiplication factor

ɛ = Fast fission factor

Lf = Fast non-leakage probability

ρ = Resonance escape probability

Lt = Thermal non-leakage probability

𝑓 = Thermal utilization factor

η = the thermal fission factor Now dont get all excited and give up, if you made it through part one, you can make it through

this. I will cover each one in detail and you will be a reactor operator in no time. On the next

page is a cheat sheet to use for our discussion. You should separate, copy, and save this for your

future.

The six factor formula and memory tool (In yellow)

Every

Little funny

Person

Loves the

Funny

Navy

The fast fission factor ɛ is the ratio of all fast neutrons produced by all fissions, to the

number of neutrons produced by thermal fission. Some fission occurs by fast neutrons. This is

particularly true for U-238 and Pu-239 which is formed by the buildup of U-238 by neutron

absorption. Remember what I said about that pesky U-238 that some times when conditions in

the nucleus are just right, that fission can occur. If the critical energy is met for the nucleus, then

it will fission. This is why U-238 is considered fissile material. A breeder reactor works on this

fast fission principle. In our PWR, this is not a large percentage of fissions, but, since we have to

account for all neutrons in our population, this factor is employed. Remember we are talking

about reactivity, any neutrons added to our core, no matter where they come from, affects

reactivity.

The fast non-leakage probability Lf This is the ratio of fast neutrons that do not leak out of the

core, to the number of fast neutrons produce by all fissions. Leakage of neutrons out of the core

must be taken into account. This factor is used to account for fast neutron leakage. When a PWR

is a power, huge amounts of neutron radiation exists in the reactor building. This is a result of

fast and thermal neutron leakage. This term leakage does not mean that there is a hole in the

core; it's just used to describe the fact that some neutrons do not interact with any of the core

materials. Instead, these neutrons are absorbed by the reactor compartment shielding.

Resonance escape probability ρ This is the ratio of neutrons that become thermalized, to the

number of fast neutrons that do not leak out of the core. This ratio is important because it tells

you how well your moderator is working and accounts for other absorption processes that are

taking place.

Thermal non-leakage probability Lt This is the ratio of thermal neutrons that do not leak out of

the core to the number of thermal neutrons. This is the same as fast non-leakage, only thermal

neutrons this time. This ratio is important because these are the neutrons that will cause fission of

U-235.

Thermal utilization factor 𝑓 This the ratio of thermal neutrons absorbed in the fuel, to the

total number of thermal neutrons. This ratio is the heart of a PWR. It tells you how well your fuel

is being supplied by thermal neutrons and how many are being sucked up by other stuff. The

equation for this is:

𝑓 = 𝜙𝑈 𝑉𝑈𝑈𝑎

𝜙𝑈𝑉𝑈+ 𝜙𝑚 𝑉𝑚 + 𝜙𝑝 𝑉𝑝+ 𝜙𝑜𝑠 𝑉𝑜𝑠𝑜𝑠𝑎

𝑝𝑎

𝑚𝑎

𝑈𝑎

where f = the thermal

utilization factor

Σ𝑎𝑈

= the total number of thermal neutrons absorbed in the fissile material, sub a,

meaning absorbed.

ϕU = the neutron flux hitting fissile material

VU = to the volume of fuel exposed to the thermal neutron flux

U, m, p, and os refers to absorption by uranium, moderator, poisons, and other stuff

Let's talk about ϕ for a second as I have not discussed this yet. It is the neutron field

strength, or flux, in a particular area. Look at it as flies buzzing about a piece of watermelon, the

flies being neutrons, and the watermelon being uranium and other stuff in the core.

The thermal fission factor η A ratio of neutrons absorbed in the fuel which cause fission to

those that are absorbed by fuel. Remember, not all thermal neutrons absorbed actually cause

fission. If the critical energy level in the nucleus is not reached, then fission will not occur. The

equation for this is:

η = 𝑁𝑈−235 𝜎𝑓

𝑈−235 𝑉𝑈−235

𝑁𝑈−235 𝜎𝑎𝑈−235 + 𝑁𝑈−238 𝜎𝑎

𝑈−238

N = to atom density (See part one)

σ = the crossection for absorption

V = to the volume of u-235

The drawing below is an example of all six factors in a full cycle. I think you can figure out the

math and ratios here. This is also a good cheat sheet for copying for future use.

Now let's talk about reactivity. It is a hard thing to get your mind around. If you look at the life

cycle above, if any one of the factors change, the reactivity in the core will also be affected.

Reactivity is a measure of that life cycle going on at any given point in time. If we pull the

control rods out with the reactor critical, then we are removing poisons (The rods) from a portion

of the core thus increasing the thermal neutron population (positive reactivity) and therefore,

reactor power goes up. As the power increases, it heats up our moderator (Coolant) which

expands and lets more thermal neutrons leak out (Negative reactivity) thus decreasing reactor

rate of power climb and stabilizing at a higher T-ave (which is the T-cold plus T-hot divided by

two)

The reactivity then goes back to 0.

The formula for reactivity (P) is as follows:

P = 𝐾𝑒𝑓𝑓 − 1

𝐾𝑒𝑓𝑓 the larger the absolute value of reactivity in the core, the further the reactor

is from being just at criticality. It is the departure from criticality either positive or negative. It is

totally dependent on what Keff is doing and thus the life cycle.

How is reactivity expressed? Well, it is kind of expressed how you want to. It is really just a

small dimensionless fraction. Below are some of the terms you may encounter for reactivity.

Δ𝐾

𝐾 Which is the change in K fraction?

1% Δ𝐾

𝐾 = .01

Δ𝐾

𝐾

pcm = .00001 Δ𝐾

𝐾 (Referred to as percent millirho)

$ (dollar sign)( enough reactivity to go prompt critical) and the 1cent sign, which just

represents which fraction your working with. But they all mean the same only differ in the way

they are expressed. Ok, I hope you were able to get a feel for what reactivity is.

Reactor period is expressed in seconds and is related to reactivity by the following equation:

τ = ℓ∗

𝑝 +

𝛽𝑒𝑓𝑓 − 𝑝

𝜆𝑒𝑓𝑓 𝑝+𝑝 where τ = reactor period in seconds. Change of power by a factor of e

ℓ∗= prompt neutron generation lifetime in seconds

Remember in part one we talked about

prompt fission neutrons and that we dont

want to be super critical with just these.

p = the reactivity expressed as a fraction or 0

𝛽𝑒𝑓𝑓 = effective delayed neutron fraction

𝑝 = rate of change in reactivity Δ𝐾/𝑘/sec

𝜆𝑒𝑓𝑓 = effective delayed neutron precursor decay constant

The first part of this equation above is for the prompt term of reactor period and the second

part is the delayed term. If control rods are withdrawn, the prompt neutrons have the greatest

affect at first. See the graph below:

Remember in part one we talked

about delayed neutrons and that they

were important for reactor control.

Well here is where they come in to

the picture. This is the fraction of all

fission neutrons born as delayed to

the total number of neutrons born.

The average time delay for delayed

neutrons < 1 minute, but can vary with the

type of fuel used. The more delay, the

more control

At initial rod movement a Prompt jump occurs 10-13 sec

Decrease in rate of climb of power is due to delayed neutron production. It makes the increasing power rise controllable

The change in reactor power (dP) = Po d 𝑡

𝜏 is dependent on the change in time to the reactor

period rate times the initial power. the transient power equation can be derived from this fact.

dP = Po d t/τ

𝑑𝑃

𝑃𝑜 = d t/τ

𝑑𝑃

𝑃0

𝑃

𝑃𝑜 = 𝑑𝑡/𝜏

𝑡

𝑡𝑜

ln𝑝𝑃

𝑃𝑜 =

1

𝜏 𝑑𝑡

𝑡

𝑡𝑜

ln P - ln Po = 1

𝜏 (t -to)

ln 𝑃

𝑃𝑜 =

𝑡

𝜏

𝑃

𝑃𝑜 = et/τ

P = Po et/τ

where: P = to the transient power level

Po = to the initial power level

e = 2.718

t = time during the reactor transient in seconds

τ = the reactor period in seconds

These two previous formulas are important in understanding what happens during life cycle

changes, the delayed neutron effects, prompt neutron effect, and reactor power changes. You

need to look at the six factors and figure in your mind what changes would occur to each if

temperature, pressure, or flow changed. Also how will control rod insertion or removal affect

each of them. And how Xenon and samarium will affect what's going on with each. This

completes part two of the neutron life cycle.